### Video Transcript

This prism has a cross section
consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are
in centimetres. Find an expression for the total
volume of the prism in cubic centimetres. Write your answer in the form π
multiplied by π₯ cubed plus π₯ squared plus π multiplied by π multiplied by π₯
cubed.

Letβs begin by reminding ourselves
what the formula is for the volume of a prism. Itβs the area of its cross section
multiplied by its length. The cross section of this prism is
a compound shape. Itβs made up of a semicircle and a
parallelogram. We do know its length though. Itβs one-half π₯.

Letβs begin by working out the area
of the cross section. Weβre going to calculate the area
of the two shapes itβs made up of. The formula for area of a circle is
ππ squared, where π is the radius. So we can find the area of a
semicircle by halving this. We can find the diameter of our
semicircle. Since opposite sides of a
parallelogram are equal in length, this means the diameter of our semicircle is
π₯. The radius is half of the length of
the diameter. So itβs a half of π₯ or π₯ over
two. The area of the semicircle is
therefore given by π multiplied by the radius squared. Thatβs π₯ over two squared all over
two.

We need to be careful when squaring
π₯ over two. We have to square both the
numerator and the denominator. That gives us π₯ squared over
four. Having a fraction within a fraction
made that last sum a little bit tricky. Instead, if we write it as a half
of π multiplied by π₯ squared over four, itβs much easier to deal with.

We multiply the numerators. Thatβs one multiplied by π
multiplied by π₯ squared. Thatβs ππ₯ squared. We then multiply the denominators,
remembering that the denominator of π will be one. Two multiplied by one multiplied by
four is eight. So the area of the semicircle is π
multiplied by π₯ squared over eight or an eighth of ππ₯ squared.

Next, we need to find the area of
the parallelogram. Thatβs found by multiplying its
base by its perpendicular height. The base of our parallelogram is π₯
and its height is π₯ plus one. So its area is π₯ multiplied by π₯
plus one. Itβs important to include the
brackets around the π₯ plus one because that reminds us that the π₯ is multiplying
both the π₯ on the inside of that bracket and by the one. π₯ multiplied by π₯ is π₯ squared
and π₯ multiplied by one is π₯. So the area of the parallelogram is
π₯ squared plus π₯.

We can see then that the total area
of the cross section is an eighth of π multiplied by π₯ squared plus another π₯
squared plus π₯. All thatβs left is to multiply this
area by the length of the prism. Thatβs one-half π₯. Weβll multiply this bracket out by
multiplying each term inside the bracket by one-half π₯. One-eighth multiplied by one-half
is one sixteenth. So we get one sixteenth of ππ₯
cubed. π₯ squared multiplied by a half π₯
is a half π₯ cubed and π₯ multiplied by a half π₯ is a half π₯ squared.

We do need to write our answer in
the form given. Notice how π₯ cubed and π₯ squared
both have a coefficient of a one-half. We can therefore factorise this
partially by taking out a factor of one-half. And a half π₯ cubed plus a half π₯
squared is the same as a half multiplied by π₯ cubed plus π₯ squared. And weβre left also with one
sixteenth of ππ₯ cubed. Comparing this to our form given in
the answer, we can see that π is equivalent to a half and π is equivalent to one
sixteenth.

And weβre done! The volume of our prism is a half
multiplied by π₯ cubed plus π₯ squared plus a sixteenth ππ₯ cubed.